ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), Supplement, 147 – 168 ALGEBRAIC THEORY OF AFFINE CURVATURE TENSORS N. BLAZIˇ C,´ P. GILKEY, S. NIKCEVIˇ C,´ AND U. SIMON Abstract. We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory. 1. Introduction Let V be a real vector space of dimension m; to simplify the discussion, we shall assume that m ≥ 4 henceforth; similar results hold in dimensions m = 2 and m = 3. In Section 2, we discuss the space of curvature operators R(V ) ⊂ ⊗2V ∗ ⊗ End(V ). These are operators with the same symmetries as those of the curvature operator of a torsion free connection on the tangent bundle of a smooth manifold. One has that R ∈ R(V ) if and only if for all x,y,z ∈ V , (1.a) R(x, y)z = −R(y, x)z and (1.b) R(x, y)z + R(y,z)x + R(z, x)y =0 . Equation (1.b) is called the first Bianchi identity. We have, see for example Strichartz [7], that 1 2 2 dim R(V )= 3 m (m − 1) . In Section 3, we discuss the space of algebraic curvature tensors a(V ) ⊂⊗4V ∗. This is the space of tensors with the same symmetries as that of the curvature tensor defined by the Levi-Civita connection of a pseudo-Riemannian metric; A ∈ a(V ) if and only if for all x,y,z,w ∈ V , (1.c) A(x,y,z,w)= −A(y,x,z,w), (1.d) A(x,y,z,w)+ A(y,z,x,w)+ A(z,x,y,w)=0, (1.e) A(x,y,z,w)= A(z,w,x,y), (1.f) A(x,y,z,w)= −A(x,y,w,z) . 2000 Mathematics Subject Classification. 53B20. Key words and phrases. algebraic curvature tensors, affine curvature tensors. N. Blaˇzi´cpassed away Monday 10 October 2005. This article is dedicated to his memory. The paper is in final form and no version of it will be submitted elsewhere. 148 N. BLAZIˇ C,´ P. GILKEY, S. NIKCEVIˇ C,´ AND U. SIMON We shall show in Theorem 3.2 that identities (1.e) and (1.f) are equivalent in the presence of identities (1.c) and (1.d). One has, see for example Strichartz [7], that: m2(m2−1) dim{a(V )} = 12 . If R ∈ R(V ), it is natural to consider the traces: ρ14(R)(x, y) := Tr z → R(z, x)y , (1.g) ρ24(R)(x, y) := Tr z → R(x, z)y , ρ34(R)(x, y) := Tr z → R(x, y)z . The identities of Equations (1.a) and (1.b) show that: ρ24(R)= −ρ14(R) and (1.h) ρ34(R)(x, y)= −ρ14(R)(x, y)+ ρ14(R)(y, x) . In Section 4, we discuss the affine curvature operators F(V ) ⊂ R(V ). These are the operators with the same symmetries as those of an affine connection without torsion; F ∈ F(V ) if and only if for all x,y,z ∈ V , (1.i) F(x, y)z = −F(y, x)z, (1.j) F(x, y)z + F(y,z)x + F(z, x)y =0, (1.k) ρ14(F)(x, y)= ρ14(F)(y, x), (1.l) ρ34(F)=0 . By Equation (1.h), Equations (1.k) and (1.l) are equivalent in the presence of Equations (1.i) and (1.j); thus these are the symmetries of the curvature operator of a torsion free, Ricci symmetric connection on the tangent bundle of a smooth manifold. Such curvature operators appear naturally as curvature operators of the induced and of the conormal connections in relative hypersurface theory; see [6]. The natural structure group of the spaces R(V ), a(V ), and F(V ) is the gen- eral linear group GL(V ). Let O(V, h·, ·i) be the orthogonal group associated to a non-degenerate symmetric bilinear form h·, ·i ∈ S2(V ∗) of signature (p, q) on V . We can use h·, ·i to raise and lower indices and define an O(V, h·, ·i) equivariant identification between ⊗4V ∗ and ⊗2V ∗ ⊗ End(V ) by means of the identity: (1.m) R(x,y,z,w)= hR(x, y)z, wi . We let r(V ) ⊂⊗4V ∗, A(V, h·, ·i) ⊂⊗2V ∗ ⊗ End(V ), f(V, h·, ·i) ⊂⊗4V ∗ be the subspaces associated to R(V ), a(V ), and F(V ), respectively; R ∈ r(V ) if and only if for all x,y,z,w ∈ V , one has R(x,y,z,w)= −R(y,x,z,w), R(x,y,z,w)+ R(y,z,x,w)+ R(z,x,y,w)=0 . ALGEBRAIC THEORY OF AFFINE CURVATURE TENSORS 149 We have A ∈ A(V, h·, ·i) if and only if for all x,y,z,w ∈ V one has: A(x, y)= −A(y, x), A(x, y)z + A(y,z)x + A(z, x)y =0, hA(x, y)z, wi = hA(z, w)x, yi, hA(x, y)z, wi = −hA(x, y)w,zi, the last two identities being equivalent in the presence of the first two. Finally F ∈ f(V, h·, ·i) if and only if for all x,y,z,w ∈ V one has: (1.n) F (x,y,z,w)= −F (y,x,z,w), (1.o) F (x,y,z,w)+ F (y,z,x,w)+ F (z,x,y,w)=0, (1.p) ρ14(F )(x, y)= ρ14(F )(y, x), (1.q) (id ⊗ Tr)F =0 . Again, identities (1.p) and (1.q) are equivalent given the identities of Equations (1.n) and (1.o). The spaces A(V, h·, ·i) and f(V, h·, ·i) depend upon the choice of the inner prod- uct; the space r(V ) does not. Thus it is convenient to keep the distinction between subspaces of ⊗2V ∗ ⊗End(V ) and ⊗4V ∗; this will play a crucial role in the proof of Theorem 4.2. The spaces R(V ), A(V ), and F(V ) are subspaces of ⊗2V ∗ ⊗End(V ); elements of these spaces will be denoted by R, A, and F, respectively, and are operator valued bilinear forms. The spaces r(V ), a(V ), and f(V ) are subspaces of ⊗4V ∗; elements of these spaces will be denoted by R, A, and F , respectively, and are quadralinear forms. We have the inclusions: A(V, h·, ·i) ⊂ F(V ) ⊂ R(V ) , a(V ) ⊂ f(V, h·, ·i) ⊂ r(V ) . 2 ∗ 4 ∗ Let {ei} be a basis for V . If ψ ∈⊗ V and if Θ ∈⊗ V , set ψij := ψ(ei,ej) and Θijkl := Θ(ei,ej ,ek,el) . Let {ei} be the associated dual basis for V ∗. Then i j i j k l ψ = ij ψij e ⊗ e and Θ= ijkl ψijkle ⊗ e ⊗ e ⊗ e . P P If h·, ·i is a non-degenerate inner product on V , let ij i (1.r) Ξij := hei,eji and j Ξ Ξjk = δk P where δ is the Kronecker symbol. One then has: ij ij ij Ξ hx, eiiej = x and Tr{ψ} = ij Ξ ψij . P P We shall decompose the natural action of GL(V ) and of O(V, h·, ·i) on the spaces R(V ), a(V ), and F(V ) as the direct sum of irreducible modules and use these decompositions to exhibit generating sets for these spaces and to derive other natural geometric properties. 150 N. BLAZIˇ C,´ P. GILKEY, S. NIKCEVIˇ C,´ AND U. SIMON Our motivation in this paper is to study affine curvature operators; as already stated above, these are the curvature operators which naturally appear as curva- ture operators of the induced and of the conormal connections in relative hyper- surface theory. Moreover, in this situation, there naturally appears a metric, the so called relative metric, which permits us to raise and lower indices. Our aim is a characterization of the affine curvature operators, arising from torsion free and Ricci symmetric connections, in the space of all curvature operators arising from torsion free connections. Via the decomposition results of Section 4, these are characterized by the vanishing of the component W3. We will study the geometric meaning of the various components in this decomposition, at least in the case of relative hypersurfaces, in a subsequent paper. 2. Curvature Operators In this section, we study operators with the same symmetries as those of a torsion free connection on the tangent bundle of a smooth manifold. 2.1. Geometric representability of curvature operators. If ∇ is a torsion free connection on the tangent bundle of a smooth manifold M, let R∇ be the associated curvature operator; if P ∈ M and if x,y,z ∈ TP M, then ∇ RP (x, y)z := ∇x∇y − ∇y∇x − ∇[x,y] z . ∇ One then has RP ∈ R(TP M) since the symmetries of Equations (1.a) and (1.b) hold. Conversely, every curvature operator is geometrically representable by an torsion free connection: Theorem 2.1. Let R ∈ R(V ) be given. Regard V as a smooth manifold in its own right. Let 0 be the origin of V and identify T0V = V . Then there exists a ∇ torsion free connection ∇ on V so that R0 = R. l Proof. Let R ∈ R(V ). Expand R(ei,ej)ek = l Rijk el relative to some basis {ei} for V . Let {xi} be the associated dual coordinates;P if e ∈ V , then e = i xi(e)ei.